Zero temperature momentum distribution of an impurity in a polaron state of one-dimensional Fermi and Tonks-Girardeau gases

Oleksandr Gamayun,Oleg Lychkovskiy,Mikhail Zvonarev

doi:10.21468/scipostphys.8.4.053

Abstract

We investigate the momentum distribution function of a single distinguishable impurity particle which formed a polaron state in a gas of either free fermions or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm determinant representation of the distribution function for the Bethe ansatz solvable model of an impurity-gas \deltaδ-function interaction potential at zero temperature, in both repulsive and attractive regimes. We deduce from this representation the fourth power decay at a large momentum, and a weakly divergent (quasi-condensate) peak at a finite momentum. We also demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the one-dimensional impenetrable anyons.

Highlights

We demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the onedimensional impenetrable anyons
In the present paper we investigate the shape of the momentum distribution function n(k, Q) of an impurity interacting with a free Fermi gas in one spatial dimension
The main result of the present paper is the Fredholm determinant representation, Eqs. (60) and (66), for the momentum distribution function, n(k, Q), of an impurity which formed a polaron state with a free Fermi gas. Using this representation we examined how the properties of the impurity depend on the strength g of the impuritygas δ-function interaction potential, and on the value of the total momentum Q of the system repulsion 1 (a) attraction: gas state (b) attraction: bound state (c)

Summary

Introduction

Non-interacting Bose and Fermi systems have markedly different momentum distribution functions at low temperature. McGuire’s solution is a special case of the Bethe ansatz solution for the Gaudin–Yang model [18,19,20], having a peculiarity that any eigenfunction can be written as a single determinant resembling the Slater determinant for the free Fermi gas [21,22,23] Such a representation, so far not available for any other interacting Bethe ansatz solvable model, enabled the derivation of an exact analytical expression for the time-dependent two-point impurity correlation function at zero [24] and arbitrary temperature [25]. Note that our result for the function (2) is valid for the impurity immersed into the Tonks–Girardeau gas This can be explained using the arguments given in the end of section 2 in Ref. Increases from 1/4 to 1/2 when γ goes from minus infinity to zero

Fredholm determinant representation in the thermodynamic limit

Determinant representation for finite N

Conclusion